Recently, functional data analysis (FDA) has received considerable attention in many fields including climatology, electromyography, and signal processing. A number of successful applications have been reported (see, e.g., Ramsay and Silverman, 1997, 2002). The basic idea behind FDA is to express observed data in the form of a function, and then draw information from a collection of functional data.
In this paper, we consider the problem of constructing functional regression models, using radial basis function networks (RBFNs) along with the technique of regularization. RBFNs combine the ideas of basis expansion and kernel smoothing methods, and may be described as a linear combination of radially symmetric nonlinear basis functions. An advantage of our approach to FDA lies in more flexibility transforming observations on each individual into a functional form.
In practice, individuals are measured at possibly differing sets of time points, so the amount of smoothness imposed on a set of discrete data could differ among subjects. Hence, in constructing functional regression models there remains the problem of how to determine the number of basis functions and an appropriate value of the regularization parameter. Cross-validation and generalized cross-validation are often referred as in the literature. However, computational effort can be enormous and there can be large variation with a tendency to undersmooth when applied to the analysis of functional data, because the selectors are repeatedly applied.
We present an information-theoretic criterion for evaluating models estimated by the method of regularization in the context of functional regression analysis. The proposed criterion is applied to choosing smoothing parameters and the number of basis functions. We analyze a set of data on climate in Canada. Bootstrap simulations were conducted to examine the performance of our modeling technique compared to the classical cross-validation method.